The coefficient of static friction between the m = 3.20-kg crate and the 35.0° incline of the figure below is 0.260. What minimum force vector F must be applied to the crate perpendicular to the incline to prevent the crate from sliding down the incline?
Wt. of crate = m * g =
3.2kg * 9.8N/kg = 31.36 N.
Fp = 31.36*sin35 = 17.99 N. = Force
parallel to the incline.
F = Fp*cos35 = 17.99*cos35 = 14.74 N.
To find the minimum force required to prevent the crate from sliding down the incline, we need to consider the forces acting on the crate in the vertical and horizontal directions.
1. Resolve the weight of the crate:
Since the crate is on an incline, we need to resolve the weight force into two components: one parallel to the incline and one perpendicular to the incline.
The weight force (W) can be resolved into two components:
- W_parallel = m * g * sin(θ)
- W_perpendicular = m * g * cos(θ)
where m = mass of the crate, g = acceleration due to gravity (9.8 m/s^2), and θ = angle of the incline (in this case, 35.0°).
2. Calculate the maximum static friction force:
The maximum static friction force (F_max) can be calculated using the coefficient of static friction (μ) and the perpendicular component of the weight force:
F_max = μ * W_perpendicular
3. Determine the minimum force required to prevent sliding:
Since we need to prevent the crate from sliding down the incline, the force perpendicular to the incline (F) must be equal to or greater than the maximum static friction force (F_max).
Therefore, F >= F_max.
Now, let's plug in the given values and calculate the minimum force:
m = 3.20 kg
θ = 35.0°
μ = 0.260
1. Calculate the weight components:
W_parallel = 3.20 kg * 9.8 m/s^2 * sin(35.0°)
W_perpendicular = 3.20 kg * 9.8 m/s^2 * cos(35.0°)
2. Calculate the maximum static friction force:
F_max = 0.260 * W_perpendicular
3. Determine the minimum force required to prevent sliding:
F_minimum = F_max
Plug in the values and calculate:
W_parallel = 3.20 kg * 9.8 m/s^2 * sin(35.0°)
= 54.04 N
W_perpendicular = 3.20 kg * 9.8 m/s^2 * cos(35.0°)
= 25.50 N
F_max = 0.260 * 25.50 N
= 6.63 N
Therefore, the minimum force vector (F) that must be applied perpendicular to the incline to prevent the crate from sliding down is 6.63 N.
To solve this problem, we need to understand the forces acting on the crate and how they relate to each other.
Let's break down the forces acting on the crate on the incline:
1. The weight of the crate (W): This force acts vertically downward and has a magnitude equal to the mass of the crate (m) multiplied by the acceleration due to gravity (g). In this case, W = m * g.
2. The normal force (N): This force acts perpendicular to the incline and opposes the vertical component of the weight. It can be calculated using N = W * cos(θ), where θ is the angle of the incline (35.0° in this case).
3. The force of static friction (f_s): This force acts parallel to the incline and opposes the motion of the crate. The maximum force of static friction can be calculated using f_s = μ_s * N, where μ_s is the coefficient of static friction.
Since we want to prevent the crate from sliding down the incline, the maximum force of static friction (f_s) must be equal to or greater than the force component pulling the crate down the incline.
Now, let's find the minimum force vector F required to prevent the crate from sliding down the incline:
1. Resolve the weight of the crate (W) into components. The component parallel to the incline is W_parallel = W * sin(θ), and the component perpendicular to the incline is W_perpendicular = W * cos(θ).
2. Calculate the force of static friction (f_s) using f_s = μ_s * N. Substitute N with W_perpendicular, since the normal force is equal to the perpendicular component of the weight. So, f_s = μ_s * W_perpendicular.
3. The minimum force vector F must be equal to or greater than f_s to prevent the crate from sliding down the incline.
In summary, to find the minimum force vector F required to prevent the crate from sliding down the incline:
1. Calculate the weight of the crate: W = m * g.
2. Resolve the weight into components: W_parallel = W * sin(θ) and W_perpendicular = W * cos(θ).
3. Calculate the force of static friction: f_s = μ_s * W_perpendicular.
4. The minimum force vector F must be equal to or greater than f_s.